Ch 09: Work and Kinetic Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 09: Work and Kinetic Energy

문제 3

9장, 문제 3

A mother has four times the mass of her young son. Both are running with the same kinetic energy. What is the ratio v_son/v_mother of their speeds?

검증된 단계별 안내

Start by recalling the formula for kinetic energy:

K = \frac{1}{2} m v^{2}

, where

K

is the kinetic energy,

m

is the mass, and

v

is the velocity.

Since the mother and son have the same kinetic energy, set their kinetic energy expressions equal to each other:

\frac{1}{2} m v_{son} v^{2} = \frac{1}{2} (4 m) v_{mother} v^{2}

, where the mother's mass is four times the son's mass.

Cancel out the common terms, such as

\frac{1}{2}

and

m

, from both sides of the equation. This simplifies to:

{v_{son}}^{2} = 4 {v_{mother}}^{2}

Take the square root of both sides to solve for the ratio of their speeds:

\frac{v_{son}}{v_{mother}} = \sqrt{4}

Simplify the square root to find the ratio:

\frac{v_{son}}{v_{mother}} = 2

. Thus, the son's speed is twice the mother's speed.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this scenario, both the mother and son have the same kinetic energy, which implies a relationship between their masses and velocities that can be analyzed to find their speed ratio.

Mass and Velocity Relationship

In physics, when two objects have the same kinetic energy, their masses and velocities are inversely related. Specifically, if one object has a greater mass, it must have a lower velocity to maintain the same kinetic energy as a lighter object. This principle is crucial for determining the speed ratio between the mother and son in the given problem.

Ratio of Speeds

The ratio of speeds (v(son)/v(mother)) can be derived from the relationship between their masses and kinetic energies. Since the mother has four times the mass of her son and both have equal kinetic energy, we can set up a proportion based on their kinetic energy equations to find the exact ratio of their speeds, illustrating how mass affects velocity in motion.